SECTION P Functions and Their Graphs 9 Section P X Domain Range = () Y A real-valued unction o a real variable Figure P Functions and Their Graphs Use unction notation to represent and evaluate a unction Find the domain and range o a unction Sketch the graph o a unction Identi dierent tpes o transormations o unctions Classi unctions and recognize combinations o unctions Functions and Function Notation A relation between two sets X and Y is a set o ordered pairs, each o the orm,, where is a member o X and is a member o Y A unction rom X to Y is a relation between X and Y that has the propert that an two ordered pairs with the same -value also have the same -value The variable is the independent variable, and the variable is the dependent variable Man real-lie situations can be modeled b unctions For instance, the area A o a circle is a unction o the circle s radius r A r A is a unction o r In this case r is the independent variable and A is the dependent variable Deinition o a Real-Valued Function o a Real Variable Let X and Y be sets o real numbers A real-valued unction o a real variable rom X to Y is a correspondence that assigns to each number in X eactl one number in Y The domain o is the set X The number is the image o under and is denoted b, which is called the value o at The range o is a subset o Y and consists o all images o numbers in X (see Figure P) FUNCTION NOTATION The word unction was irst used b Gottried Wilhelm Leibniz in 69 as a term to denote an quantit connected with a curve, such as the coordinates o a point on a curve or the slope o a curve Fort ears later, Leonhard Euler used the word unction to describe an epression made up o a variable and some constants He introduced the notation Functions can be speciied in a variet o was In this tet, however, we will concentrate primaril on unctions that are given b equations involving the dependent and independent variables For instance, the equation Equation in implicit orm deines, the dependent variable, as a unction o, the independent variable To evaluate this unction (that is, to ind the -value that corresponds to a given -value), it is convenient to isolate on the let side o the equation Equation in eplicit orm Using as the name o the unction, ou can write this equation as Function notation The original equation,, implicitl deines as a unction o When ou solve the equation or, ou are writing the equation in eplicit orm Function notation has the advantage o clearl identiing the dependent variable as while at the same time telling ou that is the independent variable and that the unction itsel is The smbol is read o Function notation allows ou to be less word Instead o asking What is the value o that corresponds to? ou can ask What is?
0 CHAPTER P Preparation or Calculus In an equation that deines a unction, the role o the variable is simpl that o a placeholder For instance, the unction given b can be described b the orm where parentheses are used instead o To evaluate, simpl place in each set o parentheses 8 7 Substitute or Simpli Simpli NOTE Although is oten used as a convenient unction name and as the independent variable, ou can use other smbols For instance, the ollowing equations all deine the same unction 7 t t t 7 gs s s 7 Function name is, independent variable is Function name is, independent variable is t Function name is g, independent variable is s EXAMPLE Evaluating a Function For the unction deined b 7, evaluate each epression a a b b c, 0 STUDY TIP In calculus, it is important to communicate clearl the domain o a unction or epression For instance, in Eample (c) the two epressions 0 and, are equivalent because 0 is ecluded rom the domain o each epression Without a stated domain restriction, the two epressions would not be equivalent Solution a a a 7 9a 7 Substitute a or Simpli b b b 7 b b 7 b b 8 Substitute b or Epand binomial Simpli c 7 7 7 7, 0 Tr It Eploration A NOTE The epression in Eample (c) is called a dierence quotient and has a special signiicance in calculus You will learn more about this in Chapter
SECTION P Functions and Their Graphs Range: 0 () = The domain o is, and the range is 0, Figure P Domain: (a) The domain o is, and the range is 0, Range () = tan π Domain (b) The domain o is all -values such that n and the range is, Figure P Range: 0 Video Editable Graph Editable Graph () = Editable Graph Domain: all real π, <, The Domain and Range o a Function The domain o a unction can be described eplicitl, or it ma be described implicitl b an equation used to deine the unction The implied domain is the set o all real numbers or which the equation is deined, whereas an eplicitl deined domain is one that is given along with the unction For eample, the unction given b, has an eplicitl deined domain given b : 5 On the other hand, the unction given b g has an implied domain that is the set : ± EXAMPLE a The domain o the unction Finding the Domain and Range o a Function is the set o all -values or which 0, which is the interval, To ind the range observe that is never negative So, the range is the interval 0,, as indicated in Figure P(a) b The domain o the tangent unction, as shown in Figure P(b), is the set o all -values such that n is an integer Domain o tangent unction The range o this unction is the set o all real numbers For a review o the characteristics o this and other trigonometric unctions, see Appendi D EXAMPLE tan n, Tr It A Function Deined b More than One Equation Determine the domain and range o the unction,, 5 Eploration A i < i Solution Because is deined or < and, the domain is the entire set o real numbers On the portion o the domain or which, the unction behaves as in Eample (a) For <, the values o are positive So, the range o the unction is the interval 0, (See Figure P) Tr It Eploration A A unction rom X to Y is one-to-one i to each -value in the range there corresponds eactl one -value in the domain For instance, the unction given in Eample (a) is one-to-one, whereas the unctions given in Eamples (b) and are not one-to-one A unction rom X to Y is onto i its range consists o all o Y
CHAPTER P Preparation or Calculus = () (, ()) () The graph o a unction Figure P5 The Graph o a Function The graph o the unction consists o all points,, where is in the domain o In Figure P5, note that the directed distance rom the -ais the directed distance rom the -ais A vertical line can intersect the graph o a unction o at most once This observation provides a convenient visual test, called the Vertical Line Test, or unctions o That is, a graph in the coordinate plane is the graph o a unction o i and onl i no vertical line intersects the graph at more than one point For eample, in Figure P6(a), ou can see that the graph does not deine as a unction o because a vertical line intersects the graph twice, whereas in Figures P6(b) and (c), the graphs do deine as a unction o (a) Not a unction o Figure P6 (b) A unction o (c) A unction o Figure P7 shows the graphs o eight basic unctions You should be able to recognize these graphs (Graphs o the other our basic trigonometric unctions are shown in Appendi D) () = () = () = () = Identit unction Squaring unction Cubing unction Square root unction () = () = () = sin () = cos π π π π π π π Absolute value unction The graphs o eight basic unctions Figure P7 Rational unction Sine unction Cosine unction
SECTION P Functions and Their Graphs EXPLORATION Writing Equations or Functions Each o the graphing utilit screens below shows the graph o one o the eight basic unctions shown on page Each screen also shows a transormation o the graph Describe the transormation Then use our description to write an equation or the transormation Transormations o Functions Some amilies o graphs have the same basic shape For eample, compare the graph o with the graphs o the our other quadratic unctions shown in Figure P8 = + 9 = (a) Vertical shit upward = ( + ) (b) Horizontal shit to the let = 9 9 Animation Animation a b 6 6 8 = = (c) Relection Animation Figure P8 = ( + ) = 5 (d) Shit let, relect, and shit upward Animation 8 0 c 5 6 6 Each o the graphs in Figure P8 is a transormation o the graph o The three basic tpes o transormations illustrated b these graphs are vertical shits, horizontal shits, and relections Function notation lends itsel well to describing transormations o graphs in the plane For instance, i is considered to be the original unction in Figure P8, the transormations shown can be represented b the ollowing equations Vertical shit up units Horizontal shit to the let units Relection about the -ais Shit let units, relect about -ais, and shit up unit d Video Basic Tpes o Transormations Original graph: Horizontal shit c units to the right: Horizontal shit c units to the let: Vertical shit c units downward: Vertical shit c units upward: Relection (about the -ais): Relection (about the -ais): Relection (about the origin): c > 0 c c c c
CHAPTER P Preparation or Calculus LEONHARD EULER (707 78) In addition to making major contributions to almost ever branch o mathematics, Euler was one o the irst to appl calculus to real-lie problems in phsics His etensive published writings include such topics as shipbuilding, acoustics, optics, astronom, mechanics, and magnetism MathBio Classiications and Combinations o Functions The modern notion o a unction is derived rom the eorts o man seventeenth- and eighteenth-centur mathematicians O particular note was Leonhard Euler, to whom we are indebted or the unction notation B the end o the eighteenth centur, mathematicians and scientists had concluded that man real-world phenomena could be represented b mathematical models taken rom a collection o unctions called elementar unctions Elementar unctions all into three categories Algebraic unctions (polnomial, radical, rational) Trigonometric unctions (sine, cosine, tangent, and so on) Eponential and logarithmic unctions You can review the trigonometric unctions in Appendi D The other nonalgebraic unctions, such as the inverse trigonometric unctions and the eponential and logarithmic unctions, are introduced in Chapter 5 The most common tpe o algebraic unction is a polnomial unction a n n a n n a a a 0, a n 0 FOR FURTHER INFORMATION For more on the histor o the concept o a unction, see the article Evolution o the Function Concept: A Brie Surve b Israel Kleiner in The College Mathematics Journal MathArticle where the positive integer n is the degree o the polnomial unction The constants a i are coeicients, with a n the leading coeicient and a 0 the constant term o the polnomial unction It is common practice to use subscript notation or coeicients o general polnomial unctions, but or polnomial unctions o low degree, the ollowing simpler orms are oten used Zeroth degree: a Constant unction First degree: a b Linear unction Second degree: a b c Quadratic unction Third degree: a b c d Cubic unction Although the graph o a nonconstant polnomial unction can have several turns, eventuall the graph will rise or all without bound as moves to the right or let Whether the graph o a n n a n n a a a 0 eventuall rises or alls can be determined b the unction s degree (odd or even) and b the leading coeicient a n, as indicated in Figure P9 Note that the dashed portions o the graphs indicate that the Leading Coeicient Test determines onl the right and let behavior o the graph a n > 0 a n < 0 a n > 0 a n < 0 Up to right Up to let Up to let Up to right Down to let Down to right Down to let Down to right Graphs o polnomial unctions o even degree The Leading Coeicient Test or polnomial unctions Figure P9 Graphs o polnomial unctions o odd degree
SECTION P Functions and Their Graphs 5 Just as a rational number can be written as the quotient o two integers, a rational unction can be written as the quotient o two polnomials Speciicall, a unction is rational i it has the orm p q, q 0 Domain o g g where p and q are polnomials Polnomial unctions and rational unctions are eamples o algebraic unctions An algebraic unction o is one that can be epressed as a inite number o sums, dierences, multiples, quotients, and radicals involving n For eample, is algebraic Functions that are not algebraic are transcendental For instance, the trigonometric unctions are transcendental Two unctions can be combined in various was to create new unctions For eample, given and g, ou can orm the unctions shown g g g g g g g g Sum Dierence Product Quotient You can combine two unctions in et another wa, called composition The resulting unction is called a composite unction g g() Domain o (g()) The domain o the composite unction Figure P0 g Deinition o Composite Function Let and g be unctions The unction given b g g is called the composite o with g The domain o g is the set o all in the domain o g such that g is in the domain o (see Figure P0) The composite o with g ma not be equal to the composite o g with EXAMPLE Finding Composite Functions Given and g cos, ind each composite unction a g b g Solution a g g Deinition o g cos Substitute cos or g cos Deinition o cos Simpli b g g Deinition o g g Substitute or cos Deinition o g Note that g g Tr It Eploration A Eploration B Eploration C Eploration D Eploration E Open Eploration
6 CHAPTER P Preparation or Calculus EXPLORATION Use a graphing utilit to graph each unction Determine whether the unction is even, odd, or neither g h 5 j 6 8 k 5 p 9 5 Describe a wa to identi a unction as odd or even b inspecting the equation In Section P, an -intercept o a graph was deined to be a point a, 0 at which the graph crosses the -ais I the graph represents a unction, the number a is a zero o In other words, the zeros o a unction are the solutions o the equation 0 For eample, the unction has a zero at because 0 In Section P ou also studied dierent tpes o smmetr In the terminolog o unctions, a unction is even i its graph is smmetric with respect to the -ais, and is odd i its graph is smmetric with respect to the origin The smmetr tests in Section P ield the ollowing test or even and odd unctions Test or Even and Odd Functions The unction is even i The unction is odd i NOTE Ecept or the constant unction 0, the graph o a unction o cannot have smmetr with respect to the -ais because it then would ail the Vertical Line Test or the graph o the unction EXAMPLE 5 Even and Odd Functions and Zeros o Functions (0, 0) (a) Odd unction (, 0) π Editable Graph (b) Even unction Editable Graph Figure P (, 0) g() = + cos π π π () = Determine whether each unction is even, odd, or neither Then ind the zeros o the unction a b g cos Solution a This unction is odd because The zeros o are ound as shown 0 Let 0 0 Factor 0,, Zeros o See Figure P(a) b This unction is even because g cos cos g cos cos The zeros o g are ound as shown cos 0 Let g 0 cos Subtract rom each side n, n is an integer Zeros o g See Figure P(b) Tr It Eploration A Eploration B Eploration C NOTE Each o the unctions in Eample 5 is either even or odd However, some unctions, such as, are neither even nor odd
SECTION P Functions and Their Graphs 7 Eercises or Section P The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem to view the complete solution o the eercise to print an enlarged cop o the graph In Eercises and, use the graphs o and g to answer the ollowing (a) Identi the domains and ranges o and g (b) Identi and g (c) For what value(s) o is g? (d) Estimate the solution(s) o (e) Estimate the solutions o g 0 g In Eercises, evaluate (i possible) the unction at the given value(s) o the independent variable Simpli the results (a) 0 (a) (b) (b) 6 (c) b (c) 5 (d) (d) 5 g 6 g (a) g0 (a) g (b) g (b) g (c) g (c) gc (d) gt (d) gt 7 cos 8 sin (a) 0 (a) (b) (b) 5 (c) (c) 9 0 In Eercises 8, ind the domain and range o the unction h g 5 5 t t sec 6 ht cot t 7 8 g g In Eercises 9-, ind the domain o the unction 9 0 g h cos sin g In Eercises 5 8, evaluate the unction as indicated Determine its domain and range 5 6 7,, (a) (b) 0 (c) (d) t,, < 0 0 > (a) (b) 0 (c) (d) s, <, (a) (b) (c) (d) b, 5 8 5, > 5 (a) (b) 0 (c) 5 (d) 0 In Eercises 9 6, sketch a graph o the unction and ind its domain and range Use a graphing utilit to veri our graph 9 0 g h 9 5 gt sin t 6 h 5 cos Writing About Concepts 7 The graph o the distance that a student drives in a 0-minute trip to school is shown in the igure Give a verbal description o characteristics o the student s drive to school Distance (in miles) 0 8 6 s (0, 0) (, ) (0, 6) (6, ) 6 8 0 Time (in minutes) t
8 CHAPTER P Preparation or Calculus Writing About Concepts (continued) 8 A student who commutes 7 miles to attend college remembers, ater driving a ew minutes, that a term paper that is due has been orgotten Driving aster than usual, the student returns home, picks up the paper, and once again starts toward school Sketch a possible graph o the student s distance rom home as a unction o time In Eercises 9, use the Vertical Line Test to determine whether is a unction o To print an enlarged cop o the graph, select the MathGraph button 9 0 0 0 In Eercises 6, determine whether is a unction o In Eercises 7 5, use the graph o unction with its graph e d 6 5 to match the 6 5 5 7 9 0 b a c, 0, > 0 5 6 0 5 () 7 5 8 5 9 50 5 6 5 g 5 Use the graph o shown in the igure to sketch the graph o each unction To print an enlarged cop o the graph, select the MathGraph button (a) (b) (c) (d) 6 9 (e) () 5 Use the graph o shown in the igure to sketch the graph o each unction To print an enlarged cop o the graph, select the MathGraph button (a) (b) (c) (d) (, ) 6 (e) () 55 Use the graph o to sketch the graph o each unction In each case, describe the transormation (a) (b) (c) 56 Speci a sequence o transormations that will ield each graph o h rom the graph o the unction sin (a) h sin (b) h sin 57 Given and g, evaluate each epression (a) g (b) g (c) g 0 (d) g (e) g () g 58 Given sin and g, evaluate each epression (a) g (b) (c) g 0 (d) g (e) g () g In Eercises 59 6, ind the composite unctions g and g What is the domain o each composite unction? Are the two composite unctions equal? 59 60 g 6 6 g g 6 Use the graphs o and g to evaluate each epression I the result is undeined, eplain wh (a) g (b) g (c) g 5 (d) g (e) g () g (, ) g cos g 7 5 g
SECTION P Functions and Their Graphs 9 6 Ripples A pebble is dropped into a calm pond, causing ripples in the orm o concentric circles The radius (in eet) o the outer ripple is given b rt 06t, where t is the time in seconds ater the pebble strikes the water The area o the circle is given b the unction Ar r Find and interpret A rt Determine the value o the constant c or each unction such that the unction its the data shown in the table 79 0 0 Think About It In Eercises 65 and 66, F g h Identi unctions or, g, and h (There are man correct answers) 65 F 66 F sin In Eercises 67 70, determine whether the unction is even, odd, or neither Use a graphing utilit to veri our result 67 68 69 cos 70 sin 80 8 8 0 0 0 8 Unde 8 0 6 0 6 Think About It In Eercises 7 and 7, ind the coordinates o a second point on the graph o a unction i the given point is on the graph and the unction is (a) even and (b) odd 7 7, 9, 7 The graphs o, g, and h are shown in the igure Decide whether each unction is even, odd, or neither h g Figure or 7 Figure or 7 7 The domain o the unction shown in the igure is 6 6 (a) Complete the graph o given that is even (b) Complete the graph o given that is odd Writing Functions In Eercises 75 78, write an equation or a unction that has the given graph 75 Line segment connecting, and 0, 5 76 Line segment connecting, and 5, 5 77 The bottom hal o the parabola 0 78 The bottom hal o the circle 6 6 6 6 8 Graphical Reasoning An electronicall controlled thermostat is programmed to lower the temperature during the night automaticall (see igure) The temperature T in degrees Celsius is given in terms o t, the time in hours on a -hour clock (a) Approimate T and T5 (b) The thermostat is reprogrammed to produce a temperature Ht Tt How does this change the temperature? Eplain (c) The thermostat is reprogrammed to produce a temperature Ht Tt How does this change the temperature? Eplain T 0 6 t 6 9 5 8 8 Water runs into a vase o height 0 centimeters at a constant rate The vase is ull ater 5 seconds Use this inormation and the shape o the vase shown to answer the questions i d is the depth o the water in centimeters and t is the time in seconds (see igure) (a) Eplain wh d is a unction o t (b) Determine the domain and range o the unction (c) Sketch a possible graph o the unction Modeling Data In Eercises 79 8, match the data with a unction rom the ollowing list (i) c (iii) h c (ii) g c (iv) r c/ d 0 cm
0 CHAPTER P Preparation or Calculus 85 Modeling Data The table shows the average numbers o acres per arm in the United States or selected ears (Source: US Department o Agriculture) Year 950 960 970 980 990 000 Acreage 97 7 6 60 (a) Plot the data where A is the acreage and t is the time in ears, with t 0 corresponding to 950 Sketch a reehand curve that approimates the data (b) Use the curve in part (a) to approimate A5 86 Automobile Aerodnamics The horsepower H required to overcome wind drag on a certain automobile is approimated b H 000 0005 009, where is the speed o the car in miles per hour (a) Use a graphing utilit to graph H (b) Rewrite the power unction so that represents the speed in kilometers per hour Find H6 87 Think About It Write the unction without using absolute value signs (For a review o absolute value, see Appendi D) 88 Writing Use a graphing utilit to graph the polnomial unctions p and p How man zeros does each unction have? Is there a cubic polnomial that has no zeros? Eplain 89 Prove that the unction is odd a n n a a 90 Prove that the unction is even a n n a n n a a 0 9 Prove that the product o two even (or two odd) unctions is even 9 Prove that the product o an odd unction and an even unction is odd 9 Volume An open bo o maimum volume is to be made rom a square piece o material centimeters on a side b cutting equal squares rom the corners and turning up the sides (see igure) 0 00 (b) Use a graphing utilit to graph the volume unction and approimate the dimensions o the bo that ield a maimum volume (c) Use the table eature o a graphing utilit to veri our answer in part (b) (The irst two rows o the table are shown) 9 Length A right triangle is ormed in the irst quadrant b the - and -aes and a line through the point, (see igure) Write the length L o the hpotenuse as a unction o Length Height, and Width Volume, V (0, ) 8 800 (, ) (, 0) 5 6 7 True or False? In Eercises 95 98, determine whether the statement is true or alse I it is alse, eplain wh or give an eample that shows it is alse 95 I a b, then a b 96 A vertical line can intersect the graph o a unction at most once 97 I or all in the domain o, then the graph o is smmetric with respect to the -ais 98 I is a unction, then a a Putnam Eam Challenge 99 Let R be the region consisting o the points o the Cartesian plane satising both and Sketch the region R and ind its area 00 Consider a polnomial with real coeicients having the propert g g or ever polnomial g with real coeicients Determine and prove the nature o These problems were composed b the Committee on the Putnam Prize Competition The Mathematical Association o America All rights reserved, (a) Write the volume V as a unction o, the length o the corner squares What is the domain o the unction?